Applications of Quadratics :
Catapults
When we model the height of a projectile shot from a
catapult as a function of the horizontal distance it has
traveled we get a quadratic function. In this activity we
use the catapults to shoot a penny into the air. We will
measure the time the penny spends in the air and the
horizontal distance it travels to determine a quadratic
equation for its motion.
For each test run, you will need one person to do each of the following:
a. Launch the catapult.
b. Watch the penny and mark the spot where it FIRST hits the ground. Mark this
spot with a piece of tape. Record the horizontal distance the penny went in
centimeters.
c. Time each run. Record how long it took the penny to reach the ground after it was
shot by the catapult in seconds.
PART I
1. Place the catapult on the ground and try a few practice shots. Make sure your
catapult does not move. Pull the catapult all the way down each time.
2. Once you start to get consistent times and distances, record three of these runs.
(Do not count runs that are not consistent with the others.) The times and
distances for the three runs you chose are:
Time
Distance
3. Calculate the average time and distance for these three runs. You will use these
average values for the future calculations.
Average Time
Average Distance
4. Fill in the blanks: At time t = 0 sec, the height of the penny is h = 0 cm. At time
t =________ the height of the penny is also h = 0 cm. In all, the penny spent
________ seconds in the air. Since half of this time the penny was rising and half
the time the penny was falling, the penny spent _______ seconds falling.
1 2
gt
2
where g is gravity g = 9.8m/sec2 =_______cm/ sec2and t is the falling time. Use
this equation to determine the highest point the penny reaches.
5. The vertical distance traveled d of an object in freefall is given by d =
d =____________
6. We wish to find a function h(x) representing the height of the penny as a function
of its horizontal distance. From the calculation in #5, we know that the ycoordinate of the vertex is _________. The x-coordinate of the vertex, will be half
of the total horizontal distance traveled. Therefore the x-coordinate of the vertex
is ________.
7. All together, the vertex of h(x) is ___________. You also know at least two other
points that lie on the graph of h(x). The point _________ lies on the graph of h(x).
8. Use your answers from #7 to determine the function h(x) and sketch a graph the
function. Recall that the standard form of a quadratic is given by
h( x ) = a ( x − h) 2 + k .
h(x) = ___________________________
PART II
9. The axis of symmetry for the quadratic equation found in #8 is x =_________.
The x-intercepts of the quadratic equation are ____________________. Label
these on the graph sketched in #8.
10. Calculate how far away the catapult should be from the wall in order for the
penny to hit the wall at the vertex. Justify your reasoning. (Hint: Think about your
answers in #9.)
The catapult should be placed __________ centimeters from the wall.
11. Measure the height of the vertex on the wall and mark with a piece of tape. Test
your calculation from #10. How close is the penny to the vertex?
PART III
12. Place the catapult on a raised surface. (You can use your desk, a stack of books,
ect.) Measure how high this surface is from the floor. The distance is
__________ cm.
13. Give a sketch of the quadratic function describing the motion of the penny when
the catapult is place on this raised surface. Label the y-intercept.
14. Using what you learned about translations, write the new equation for your
quadratic function h(x).
h(x) = ___________________________
15. The vertex of h(x) is ____________. Therefore the highest the penny will reach is
___________ cm.
16. Calculate where your penny should land when it is shot from this height. Mark
this spot on the floor. The penny should land _____________ cm away.
17. Shoot the catapult three times and measure the horizontal distances.
Shot Number
Horizontal Distance
Difference between actual
distance and distance calculated
in #10
(take absolute value)
1
2
3
18. How good was the estimated distance given in #16? If you do not get a good
approximation, give some possible (physical) explanations for why your
calculation did not agree with your results.